From: AAAI-99 Proceedings. Copyright © 1999, AAAI (www.aaai.org). All rights reserved.
Total Order Planning is More Efficient than we Thought
Vincent VIDAL, Pierre RÉGNIER
Department of “Raisonnement et décision”, IRIT, Paul Sabatier University
118 route de Narbonne
31062 Toulouse cedex, FRANCE
E-mail: {vvidal, regnier}@irit.fr
Abstract”
In this paper, we present VVPLAN, a planner based on a
classical state space search algorithm. The language used
for domain and problem representation is ADL (Pednault
1989). We have compared VVPLAN to UCPOP (Penberthy
and Weld 1992)(Weld 1994), a planner that admits the same
representation language. Our experiments prove that such
an algorithm is often more efficient than a planner based on
a search in the space of partial plans. This result is achieved
as soon as we introduce in VVPLAN’s algorithm a loop test
relating to previously visited states. In particular domains,
VVPLAN can also outperform IPP (Koehler et al. 1997),
which makes a planning graph analysis as GRAPHPLAN.
We present here the details of our comparison with UCPOP,
the results we obtain and our conclusions.
1 Introduction
1.1 Preliminaries
Planning has represented an important part of Artificial
Intelligence for almost forty years and has received
numerous developments. To solve more and more difficult
problems, especially in robotics and productics domains, it
has improved its formalism of representation and its
algorithms. So, various planners have been developed.
Among them, the most recent are commonly considered as
more efficient than the eldest. The most employed
techniques are search in the space of states, search in the
space of partial plans and planning graph analysis. The
first one — the eldest, (Fikes and Nilsson 1971) — is
based on backward or forward state space search and the
second one — more recent, (Chapman 1987) — on
refinement strategies in space of partial plans
(Kambhampati and Srivastava 1995), (Kambhampati
1997). Recently, the planners GRAPHPLAN (Blum and
Furst 1997), then IPP (Koehler et al. 1997) have provided
excellent results analyzing planning graphs.
We detail here an experimental study about VVPLAN, a
forward state space planner for the ADL language. The
numerous results we have achieved allow us to discuss the
common opinion of the planning community “total-order
planners are less efficient than partial-order planners”.
Copyright © 1999, American Association for Artificial Intelligence
(www.aaai.org). All rights reserved.
Indeed, the overwhelming majority of works that support
that point of view does not take into account the
knowledge about the current state of the world (which is
realized planning forward). A total order planner provides
good performances as soon as we use this knowledge to
prune the search tree. These results complete other works
as those of (Veloso and Blythe 1994), (Bacchus and
Kabanza 1995) and HSP (www.ldc.usb.ve/~hector), (Bonet,
Loerincs and Geffner 1997), demonstrating that state space
search can be efficient in a lot of problems.
1.2 The current opinion
At present, the widespread belief in the planning
community is that planning in the space of partial plans is
more efficient than planning in the space of totally ordered
plans or than planning in the space of states. So, in several
important papers, we can quote numerous paragraphs
where the authors give their opinion about total order and
partial order planning according to this belief:
(Penberthy and Weld 1992): Since consensus suggests
that partial order planning is preferable to total order
approaches (…), we pondered the void in the space of
rigorous planners.
(Kambhampati and Chen 1993): In contrast, the
common wisdom in the planning community (…), has
held that search in the space of plans, especially in the
space of partially ordered plans provides a more flexible
and efficient means of plan generation.
(Kambhampati 1995): The conventional wisdom of the
planning community, supported to a large extent by the
recent analytical and empirical studies (…), holds that
searching in the space of plans provides a more flexible
and efficient framework for planning.
(Blum and Furst 1995): It may seem puzzling that an
extra level of commitment would lead to a fast planner,
especially given the success enjoyed by leastcommitment planners (…).
(Bacchus and Kabanza 1995): The choice between the
various search spaces has been the subject of much
recent inquiry (…), with current consensus seemingly
converging on the space of partially ordered plans (…).
These opinions are fundamentally based on several
experimental studies that conclude to the superiority of
planning in the space of partially ordered plans on
planning in the space of totally ordered plans: (Minton,
Bresina and Drummond 1991), (Minton et al. 1992),
(Minton, Bresina and Drummond 1994) and (Barrett and
Weld 1994). Generally, those who quote these papers
wrongly assimilate total order approaches and state space
planning. In spite of this almost unanimous point of view
(about the supposed efficiency of the partial order
approach), two other papers — not well known and
opposed to the precedent ones — really compare partial
order planning and state space planning: (Veloso and
Blythe 1994) demonstrate that some problems of particular
domains can be solved more efficiently planning backward
in the space of states than using partial order approaches.
(Bacchus and Kabanza 1995) also compare, to its best
advantage, the planner TLPLAN (forward state space
planner, special language to express search control
knowledge) to the planner SNLP (partial order planner).
Systematic comparisons between different planners
begin to be dealt in controlled competitions — the first one
has been organized during the last congress AIPS’98
(www.cs.yale.edu/HTML/YALE/CS/HyPlans/mcdermott.html). These
experiments demonstrate the superiority of the
GRAPHPLAN-like algorithms. This last generation of
planners generally outperforms the conventional methods
in most cases.
The experimental studies we detail here clearly
demonstrate that the question is far from being solved, at
least concerning the comparison between partial plans
planning and planning in the space of states. First of all, we
are going to sum up the papers we have quoted before:
(Minton, Bresina and Drummond 1994) — revise and
complete (Minton, Bresina and Drummond 1991), (Minton
et al. 1992) — (Barrett and Weld 1994) which give a
complementary and interesting viewpoint about the
classification of planning domains, and (Veloso and Blythe
1994). Afterwards (cf. § 3), we will demonstrate that
comparing UCPOP (Penberthy and Weld 1992), which is a
reference for partial order planning, with VVPLAN, a
forward state space planner, tests prove superior
performances for the latter as soon as we prune the search
tree thanks to the knowledge on already visited states.
These results, which are in opposition to the common point
of view, can be easily explained because most of the
previous studies do not take into account this knowledge.
2 Main related works
(Minton, Bresina and Drummond 1994) compare a planner
that searches in a space of partially ordered plans and a
planner that searches in a space of totally ordered plans to
determine the factors influencing performances and their
respective importance (dimension of the search space,
time cost of a refinement per plan, role of search strategies
and heuristics, description language…). These algorithms
are tested on a large set of randomly generated problems in
the blocks world domain for various versions and
heuristics. Performance criteria are the number of
developed nodes and CPU time. These works demonstrate
that in the blocks world domain and with or without
domain independent heuristics, the partial-order planner
UA is more efficient than the total-order planner TO.
(Barrett and Weld 1994) compare three planning
algorithms which are tested on several artificial domains:
POCL (space of partially ordered plans), TOCL (space of
totally ordered plans, pre-ordering tractability refinements)
and TOPI (space of totally ordered plans, adds operators to
the head of the plan and is equivalent to a backward state
space planner). They propose an extension of Korf’s
taxonomy of subgoal collections (Korf 1987) for planning
in the space of partial plans1. Domains are built in order to
present independent, trivially serializable or laboriously
serializable subgoals for the different algorithms.
Performances are estimated in CPU time, according to the
number of subgoals. In all the domains, POCL and TOCL
perform significantly better than TOPI, POCL being itself
more efficient than TOCL. Those results support the
conclusions of (Minton, Bresina and Drummond 1994).
Afterwards, POCL and TOCL are tested on the fairly
complex “Tyre world” domain which can not be solved in
its initial version neither by POCL, nor by TOCL because
of the laboriously serializable subgoals. If subgoals are
ordered to make them serializable, the problem is then
solved by POCL and TOCL (POCL being much more
efficient). TOPI always fails.
(Veloso and Blythe 1994) compare SNLP (planning in a
space of partially ordered plans) with PRODIGY (planning
in a space of totally ordered plans, computation of the
current state of the world to control the search). They
define the linkability2 property in order to predict the more
efficient algorithm for solving a particular problem. SNLP
is quasi identical to POCL, and PRODIGY is based on an
algorithm similar to TOPI (backward planning in a space
of states with possible computation of the current state of
the world to control the search). These algorithms are
tested on randomly generated problems from several
artificial domains specially built to present, or not, the
linkability property. Performances are estimated in CPU
time, according to the number of subgoals. This method is
similar to those of (Barrett and Weld 1994), except for the
domain property. This time, results are favorable to total
order planning in accordance with the selected criteria
(CPU time and algorithms linkability property for the
tested domains). In all cases, PRODIGY outperforms
SNLP. This conclusion is absolutely contradictory to the
two previous studies. The main reason is the use of a
simulated world state which allows a backtrack reduction
on the studied domains. In all the tested domains, goals are
1 For example, in a given domain, problems may have trivially
serializable subgoals for an algorithm and laboriously serializable
for another one.
2 This property is related to the order employed to post interval
protection constraints in partial order planning: when this order is
wrongly chosen, the algorithm must backtrack and works less
efficiently.
trivially linkable for PRODIGY (solved in polynomial time
according to the number of subgoals) and laboriously
linkable for SNLP (solved in exponential time according to
the number of subgoals). This property is therefore rather
favorable to total order planning. These conclusions widely
moderate those of the two previous papers.
See also (Kambhampati 1994) and (Kambhampati and
Srivastava 1995), (Kambhampati and al. 1996) which
detail a complete comparison of different plan space
planning algorithms. All these different works show the
following points:
x Nor the superiority of partial order on total order
planning is proved neither the contrary. We can find
domains that take advantage of both techniques.
x An ambiguity remains concerning state space planning:
an algorithm that searches in a space of totally ordered
plans like TOPI is theoretically equivalent to a backward
state space search algorithm, but it does not use the
knowledge on the current state of the world to control
the search. When this is done, as with PRODIGY
(Veloso and Blythe 1994) or TLPLAN (Bacchus and
Kabanza 1995), results are widely modified.
x Nor the linkability neither the serialization are sufficient
criteria to evaluate a planner. Some other criteria should
be employed and then their respective influences should
be studied.
x No comparison between partial order planning and
forward state space planning has been done using
already produced states to prune the search tree. The
main interest of (Bacchus and Kabanza 1995) is the
description of a special language employed to control
the search, using domain dependent knowledge.
3 Comparison UCPOP / VVPLAN
3.1 Language and equipment
We have compared the UCPOP planner (version 4.1) with
VVPLAN, a planner based on a state space search
algorithm. Both planners used ADL to represent problems.
We performed our tests in identical conditions: same
programming language (CMU Common Lisp) and same
equipment (PC with linux, CPU Intel Pentium 200 Mhz).
Problems we have used are the 39 classical ones supplied
with UCPOP. In order to reduce the influence of the
system (particularly the Lisp garbage collecting influence),
each has been tested several times: 100 times concerning
problems solved in less than 0,1 seconds, 10 times
concerning problems solved in 1 or 2 seconds, and only
one, concerning the others. These tests clearly point out the
greatest efficiency of VVPLAN as soon as we introduce a
test based on memoization (called state loop control).
heuristic. VVPLAN is built on a classical forward state
space search algorithm. It accepts as input the current state
of the world, the goal and the current plan. It begins with
the initial state of the problem and the empty plan. The
current plan is modified during the whole planning
process. It is a totally ordered operator sequence. Three
versions of this algorithm have been developed.
Refine-state-space (S: state, g: goal, P: plan)
1. Termination check:
- if the goal g belongs to the state S, return
P (P is a solution)
2. Action selection:
- choose an action a using an operator in the
set of operators, so that its preconditions
belong to the state S
- if no action can be applied to S, return fail
3. Action application:
- apply the action a to S to reach a state S'
4. Recursive invocation:
- Refine-state-space (S', g, P + ¢ a ² 3)
Figure 1: The VVPLAN algorithm.
x VVPLAN-breadth-first without state loop control: is a
classical breadth-first search algorithm; during steps 2
and 3 of the algorithm (cf. figure 1), all the possible
states are created, applying all the possible actions to the
current state of the world. Afterwards, those states are
placed in a FIFO list. Search is carried on using the first
state of the list as current state.
x VVPLAN-breadth-first with state loop control: this
algorithm is similar to the previous one except for the
further test: when the current state has been yet visited
or is already situated in the FIFO list, it is not added to
the latter. Even though this test seems to be
expensive (numerous states must be memorized and the
more numerous the created states are, the more long the
test is), our implementation (numerical encoding of the
states, hash-coding tables) produces good performances.
x VVPLAN-A*: is a classical A* heuristic search
algorithm. The difference with the previous one relies
on the order used to pick up the states from the list. To
order the states, we do not use a FIFO method but the
minimum value of a function f = g + h which adds: the
cost g (length in number of steps) of the already realized
path from the initial state to the current one, and the
estimated value, using a heuristic4 function h, of the
length of the remaining path (in number of steps) from
the current state to the final one.
This algorithm returns the shortest plan if and only if the
heuristic function h minimizes the real cost of the
remaining path. When several states have the same value
for f, we use the optimistic interpretation: we pick up the
3 The + operation joins two operators sequences: if P = ¢ a , …,
1
an ², then we have: P + ¢ an+1 ² = ¢ a1, …, an, an+1 ².
3.2 The compared algorithms
4 We use a domain independent heuristic h: the maximization of
The search algorithm of UCPOP which is our reference
remains the same during all the tests: an A* algorithm in
the partial plans space using a domain independent
the number of solved subgoals in each state. This heuristic is
generally admissible because in most of the tested domains, an
action never adds at once more than one subgoal.
state with the greatest value for g i.e. both the most distant
state from the initial one (g maximal) and the probably
closest state to the goal (h minimal).
3.3 Description of the tests
They concern 39 different classical problems from 15
various domains, which are issued from numerous planners
(STRIPS, PRODIGY, SNLP…). They have been adapted
and improved thanks to the ADL language to be used by
UCPOP. Those problems are performed using the three
successive versions of VVPLAN. Results are estimated in
CPU time and in the number of created and developed
nodes. Search fails when it exceeds 100,000 created nodes.
3.4 Results
Comparison UCPOP / VVPLAN-breadth-first without
state loop control. As we can see in figure 2, this first
comparison gives the advantage to UCPOP. It solves 32
problems among the 39 ones (82 %) compared with only
26 for VVPLAN (66 %). Concerning these 26 problems
(solved by the two planners), 12 (46 %) are performed
faster by UCPOP; among the 32 problems solved by
UCPOP, 18 (56 %) are solved faster by UCPOP. The
difference is not so important but VVPLAN fails on 13
problems compared to 7 for UCPOP. Consequently, when
VVPLAN uses the breadth-first algorithm without state
loop control, these results point out UCPOP’s superiority.
These results seem to confirm those of (Minton, Bresina,
Drummond 1994), but 14 problems are solved faster by
VVPLAN. This points out that the domains used in these
studies are not different enough and are most often in favor
of partial order planning.
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Comparison UCPOP / VVPLAN-breadth-first with
state loop control. When we add state loop control in
VVPLAN’s breadth-first search algorithm, results become
totally different (see figure 4):
x All the 39 problems are now solved by VVPLAN, always
compared with 32 (82%) problems for UCPOP. 23
problems among the 32 problems solved by the two
planners are solved faster by VVPLAN (72 %, 9 more
than with the first version of VVPLAN). On the average,
VVPLAN solves those 32 problems 7 times faster than
UCPOP (UCPOP 3.57 sec. against VVPLAN 0.48 sec.).
x VVPLAN creates far less nodes than UCPOP. Figure 5
(ratio of the created nodes: nUCPOP / (nUCPOP + nVVPLAN)) is
similar to the CPU time graph. For 22 problems among
the 32 ones solved by the two planners (69 %), UCPOP
creates more nodes than VVPLAN.
x UCPOP also develops more nodes than VVPLAN in 30
of the 32 problems (93 %).
1
0.8
1
0.6
0.8
0.4
0.6
0.2
0
303126 4 6 1 2 1713181020 3 21123916 9 3211 5 3635282737342538332914 7 8 1519222324
0.4
Figure 4: UCPOP vs. VVPLAN-breadth-first with
state loop control (CPU time).
0.2
0
6 9 12 13 21 26 39 30 4 20 1 11 3 38 2 10 35 17 31 18 16 37 33 36 25 32 5 28 27 34 14 29
Figure 2: UCPOP / VVPLAN-breadth-first without state
loop control (CPU time). X-axis shows the tested problems
(see figure 3 for the numbering of problems), and Y-axis the
relative value between the CPU time of the two planners: tUCPOP /
(tVVPLAN + tUCPOP). Problems appear on the X-axis in the increasing
order of the values, so as to obtain a usable curve. This order
can vary from one curve to the other. The horizontal line (at 0.5)
separates the zone in which UCPOP is more efficient (points
bellow the line) from the zone in which VVPLAN is more efficient
(points over the line). Problems at the left of the first vertical line
(if there are some) remain unsolved by VVPLAN, and the ones at
the right of the second line (if there are some) are not solved by
UCPOP.
1
0.8
0.6
0.4
0.2
0
30 4 263113 1 2 101239113632 9 35 3 202721 6 5 3717342818163825332914 7 8 1519222324
Figure 5: UCPOP vs. VVPLAN-breadth-first with
state loop control (created nodes).
When these results are compared with the ones made
without state loop control, we can point out that:
x On one hand, state space planning often develops a larger
search space than partial order planning, which denies
the current opinion.
x On the other hand, the average branching factor (number
of created nodes / number of developed nodes) is higher
for VVPLAN (6.38 against 1.44 for UCPOP), which
confirms the established opinion.
x The difference of performance between these two
versions of VVPLAN — without and with state loop
control — demonstrates the interest of this test. The
influence of the branching factor on the size of the
search graph is minimized due to the tremendous
number of pruned nodes.
Researchers using the results of (Barrett and Weld 1994)
and (Minton, Bresina and Drummond 1994) to affirm that
partial order planning is more efficient than state space
planning make two mistakes:
x A total order planning algorithm as TOPI is not really
equivalent to a state space planning algorithm. TOPI is
indeed a very simple algorithm, without any
improvement like state loop control. This kind of test
can only be made by an algorithm which really
computes states like VVPLAN. Even though this control
seems to be expensive (in time and memory), it leads to
really good performances thanks to the pruning of the
search tree.
x (Barrett and Weld 1994) always point out that they have
never obtained good performances for total order
planning but only on their domains. Though they used a
lot of different problems, it is not sufficient to prove that
TOPI is always less efficient.
If we study the nature of the problems, we can notice:
x UCPOP’s unsolved problems have a long plan-solution:
on the average 12.6 actions compared with 5.2 actions
for solved problems. Unsolved problems are complex
ones and include a lot of operators and facts (problems
“fixit”, “fixb”, “move-boxes”), or are problems with
laboriously serializable subgoals (problem “hanoi4”: the
only possible serialization of subgoals is from the
biggest disc to the smallest).
x The problem “fixit” is the famous one of the “Tyre
World” domain (see § 2). According to (Barrett and
Weld 1994), this problem can only be solved by a partial
order planner like POCL or by a total order planner like
TOCL (if subgoals are proposed in a serializable order)
but never by a total order planner (assimilated to a state
space planner) as TOPI. (Barrett and Weld 1994, p. 99)
notice: “We took as our challenge, the problem of
rendering this problem tractable”. As it is demonstrated
by our results, UCPOP does not solve this problem in its
original form (without serialized subgoals) even though
VVPLAN solves it in almost 6 sec, whatever is the
subgoals order. VVPLAN creates 7,153 nodes to solve
it, and UCPOP fails with more than 100,000 nodes.
x UCPOP’s success on problems “sched-test1a”, “sched-
test2a” and “rat-insulin” can be explained because these
problems have a lot of operators with few preconditions.
This characteristic leads to an important branching
factor for VVPLAN, but to a very little one for UCPOP
(there are few constraints among actions). Partial plans
developed by UCPOP have a lot of parallel actions.
Thus, UCPOP’s search space is far less important than
VVPLAN’s. It is typically the kind of problems that are
solved faster by a partial order planner.
x Among the problems solved more efficiently by UCPOP,
4 are sub-problems of more complex ones that are not
solved by UCPOP: “fix1”, “fix2” and “fix4” are subproblems of “fixit” and “fixa” is a sub-problem of
“fixb”. Sub-problem of a more complex one means a
problem with an initial state to be reached in order to
solve the complex problem, and with a goal to reach
before solving the one of the complex problem. Despite
the fact that UCPOP is efficient on easy problems of
certain domains, it works laboriously on more complex
problems in the same domains.
Comparison UCPOP / VVPLAN-A*. VVPLAN is now
improved using an A* algorithm (with state loop control)
and a classical domain-independent heuristic function: the
maximization of the number of solved subgoals (in each
state). The results we achieve are slightly better than the
previous ones, even though it does not clearly appear on
the figure 6 (CPU time). VVPLAN solves all the 39
problems, and UCPOP still solves 32 problems (82 %). 24
problems (one more than previously) are solved faster by
VVPLAN (75 %) and the average CPU time (concerning
these 32 problems) is now 0.46 sec. (against 0.48 sec.
before). Furthermore, VVPLAN creates and develops
slightly less nodes than in the precedent tests.
This curve does not appear to be very significant because it
does not point out the improvement realized using the A*
algorithm. This improvement appears if we take into
account the 39 problems (cf. figure 7).
1
0.8
0.6
0.4
0.2
0
3031 2 26 6 5 1017 4 1320 1 211836 9 11123916 3 2827323734352533382914 7 8 1519222324
Figure 6: UCPOP / VVPLAN-A* with state loop control
(CPU time).
We notice that:
x 31 problems (79 %) are solved faster by VVPLAN.
x The average CPU time (which was 2.10 sec. with
VVPLAN-breadth-first) is now 1 sec. So, the time for
evaluating the heuristic function does not penalize the
overall performances. A* algorithm gives significantly
better results on the 7 most difficult problems (which
remain unsolved by UCPOP). Particularly, the CPU time
needed by the two most difficult problems is now
divided by 7 (problem “move-boxes”) and by 6
(problem “prodigy-p22”). UCPOP’s total average CPU
time (109 sec.) is not really significant because it fails
on 7 problems. However, this average CPU time is more
than100 times higher than VVPLAN’s.
x VVPLAN-A* creates (and develops) almost 30 % less
nodes than VVPLAN-breadth-first when considering the
39 problems. This difference was nearly zero when
considering the only 32 problems solved by the two
planners. UCPOP creates 28 times more nodes and
develops 98 times more nodes than VVPLAN-A*. This
clearly points out VVPLAN-A*’s superiority on the
performed examples.
UCPOP
CPU Time
(sec.)
Created
nodes
Developed
nodes
32 pbs.
39 pbs.
32 pbs.
39 pbs.
32 pbs.
39 pbs.
3.57
> 109
2,286
> 23,294
1,577
> 15,337
VVPLAN
state loop control
0.48
2.10
511
1,192
80
239
A*
0.46
1.00
494
845
76
156
Figure 7: Results for all the problems.
5 Conclusion
In this paper, we have detailed an experimental
comparison between VVPLAN, a forward state space
planner, and the planner UCPOP. The results of these tests,
allowed by the introduction of state loop control, show that
VVPLAN outperforms UCPOP in most of the problems.
We also compared VVPLAN to IPP (results are not
given here because of the lack of space). These tests
demonstrate that some classes of problems can be solved
more efficiently with a state space planner, giving an
optimal solution. The efficiency of GRAPHPLAN-like
planners is due to the internal qualities of this kind of
algorithm, but this result is achieved against the quality of
the solution. The latter is no more optimal in the number of
actions, but in the number of time steps (a time step can
hold several actions). We can suppose that problems of
some domains, with no parallelism, a restricted branching
factor and numerous redundant states, can be solved faster
using a state space planner.
Finally, our feeling is that no planner outperforms all the
others, in every domain. The differences of performance
we observed seem to come essentially from some
characteristics of the domains. It could be interesting to
systematically try to characterize the numerous properties
of domains (serialization of subgoals, linkability, number
of operators, number of preconditions, effects and
interactions…) to associate to each domain the probably
most efficient algorithm.
Acknowledgements
This research was greatly improved by discussions and
comments by E. Jacopin and S. Souville. We also thank the
anonymous reviewers of this paper.
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